Another important high dimensional system of coupled ordinary differential
equations is an ensemble of N all-to-all coupled phase
oscillators [9] .
It is defined as

dφ​k / dt = ω​k + ε / N Σ​j sin( φ​j - φ​k )

The natural frequencies ω​i of each oscillator follow
some distribution and ε is the coupling strength. We
choose here a Lorentzian distribution for ω​i. Interestingly
a phase transition can be observed if the coupling strength exceeds a critical
value. Above this value synchronization sets in and some of the oscillators
oscillate with the same frequency despite their different natural frequencies.
The transition is also called Kuramoto transition. Its behavior can be analyzed
by employing the mean field of the phase

Z = K ei Θ = 1 / N Σ​kei φ​k

The definition of the system function is now a bit more complex since we
also need to store the individual frequencies of each oscillator.

Now, we do several integrations for different values of ε
and record Z. The result nicely confirms the analytical
result of the phase transition, i.e. in our example the standard deviation
of the Lorentzian is 1 such that the transition will be observed at ε =
2.